József Sándor GEOMETRIC THEOREMS EQUATIONS AND ARITHMETIC .
József SándorGEOMETRIC THEOREMS, DIOPHANTINEEQUATIONS, AND ARITHMETIC C (MB/MC)(sin u / sin v)1/x 1/y 1/zZ(n) is the smallest integer msuch that 1 2 m is divisible by n*************************************American Research PressRehoboth2002
József SándorDEPARTMENT OF MATHEMATICSBABEŞ-BOLYAI UNIVERSITY3400 CLUJ-NAPOCA, ROMANIAGeometric Theorems, Diophantine Equations, andArithmetic FunctionsAmerican Research PressRehoboth2002
This book can be ordered in microfilm format from:Books on DemandProQuest Information and Learning(University of Microfilm International)300 N. Zeeb RoadP.O. Box 1346, Ann ArborMI 48106-1346, USATel.: 1-800-521-0600 (Customer Service)http://wwwlib.umi.com/bod/Copyright 2002 by American Research PressRehoboth, Box 141NM 87322, USAMore books online can be downloaded from:http://www.gallup.unm.edu/ smarandache/eBook-otherformats.htmReferents:A. Bege, Babeş-Bolyai Univ., Cluj, Romania;K. Atanassov, Bulg. Acad. of Sci., Sofia,Bulgaria;V.E.S. Szabó, Technical Univ. of Budapest,Budapest, Hungary.ISBN: 1-931233-51-9Standard Address Number 297-5092Printed in the United States of America
”.It is just this, which gives the higher arithmetic that magical charm which has madeit the favourite science of the greatest mathematicians, not to mention its inexhaustiblewealth, wherein it so greatly surpasses other parts of mathematics.”(K.F. Gauss, Disquisitiones arithmeticae, Göttingen, 1801)1
PrefaceThis book contains short notes or articles, as well as studies on several topics ofGeometry and Number theory. The material is divided into five chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbersand functions; and Some irrationality results. Chapter 1 deals essentially with geometricinequalities for the remarkable elements of triangles or tetrahedrons. Other themes havean arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2includes various diophantine equations, some of which are treatable by elementary methods; others are partial solutions of certain unsolved problems. An important method isbased on the famous Euler-Bell-Kalmár lemma, with many applications. Article 20 maybe considered also as an introduction to Chapter 3 on Arithmetic functions. Here manypapers study the famous Smarandache function, the source of inspiration of so manymathematicians or scientists working in other fields. The author has discovered variousgeneralizations, extensions, or analogues functions. Other topics are connected to the composition of arithmetic functions, arithmetic functions at factorials, Dedekind’s or Pillai’sfunctions, as well as semigroup-valued multiplicative functions. Chapter 4 discusses certain divisibility problems or questions related especially to the sequence of prime numbers.The author has solved various conjectures by Smarandache, Bencze, Russo etc.; see especially articles 4,5,7,8,9,10. Finally, Chapter 5 studies certain irrationality criteria; some ofthem giving interesting results on series involving the Smarandache function. Article 3.13(i.e. article 13 in Chapter 3) is concluded also with a theorem of irrationality on a dualof the pseudo-Smarandache function.A considerable proportion of the notes appearing here have been earlier published in2
journals in Romania or Hungary (many written in Hungarian or Romanian).We have corrected and updated these English versions. Some papers appeared alreadyin the Smarandache Notions Journal, or are under publication (see Final References).The book is concluded with an author index focused on articles (and not pages), wherethe same author may appear more times.Finally, I wish to express my warmest gratitude to a number of persons and organizations from whom I received valuable advice or support in the preparation of this material.These are the Mathematics Department of the Babeş-Bolyai University, the Domus Hungarica Foundation of Budapest, the Sapientia Foundation of Cluj and also ProfessorsM.L. Perez, B. Crstici, K. Atanassov, P. Haukkanen, F. Luca, L. Panaitopol, R. Sivaramakrishnan, M. Bencze, Gy. Berger, L. Tóth, V.E.S. Szabó, D.M. Milošević and the lateD.S. Mitrinović. My appreciation is due also to American Research Press of Rehoboth forefficient handling of this publication.József Sándor3
ContentsPreface2Chapter 1. Geometric theorems81On Smarandache’s Podaire Theorem . . . . . . . . . . . . . . . . . . . . .92On a Generalized Bisector Theorem . . . . . . . . . . . . . . . . . . . . . . 113Some inequalities for the elements of a triangle . . . . . . . . . . . . . . . . 134On a geometric inequality for the medians, bisectors and simedians of anangle of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165On Emmerich’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 196On a geometric inequality of Arslanagić and Milošević . . . . . . . . . . . . 237A note on the Erdös-Mordell inequality for tetrahedrons . . . . . . . . . . 258On certain inequalities for the distances of a point to the vertices and thesides of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279On certain constants in the geometry of equilateral triangle . . . . . . . . . 3510The area of a Pythagorean triangle, as a perfect power . . . . . . . . . . . 3911On Heron Triangles, III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212An arithmetic problem in geometry . . . . . . . . . . . . . . . . . . . . . . 53Chapter 2. Diophantine equations1 111On the equation in integers . .x yz1112On the equation 2 2 2 in integersxyza bca b3On the equations and x ydx y456. . . . . . . . . . . . . . . . . . 57. . . . . . . . . . . . . . . . . . 59c. . . . . . . . . . . . . . . . . 62z
6The Diophantine equation xn y n xp y q z (where p q n) . . . . . . . 641111 . . . . . . . . . . . 65On the diophantine equationx1 x2xnxn 1On the diophantine equation x1 ! x2 ! . . . xn ! xn 1 ! . . . . . . . . . . 687The diophantine equation xy z 2 1 . . . . . . . . . . . . . . . . . . . . . 708A note on the equation y 2 x3 1 . . . . . . . . . . . . . . . . . . . . . . 729On the equation x3 y 2 z 3 . . . . . . . . . . . . . . . . . . . . . . . . . 7510On the sum of two cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7711On an inhomogeneous diophantine equation of degree 3 . . . . . . . . . . . 8012On two equal sums of mth powers . . . . . . . . . . . . . . . . . . . . . . . 83nX(x k)m y m 1 . . . . . . . . . . . . . . . . . . . . . 87On the equation4513k 114On the diophantine equation 3x 3y 6z15On the diophantine equation 4x 18y 22z . . . . . . . . . . . . . . . . . 9116On certain exponential diophantine equations . . . . . . . . . . . . . . . . 9317On a diophantine equation involving arctangents . . . . . . . . . . . . . . . 9618A sum equal to a product . . . . . . . . . . . . . . . . . . . . . . . . . . . 10119On certain equations involving n! . . . . . . . . . . . . . . . . . . . . . . . 10320On certain diophantine equations for particular arithmetic functions . . . . 10821On the diophantine equation a2 b2 100a b . . . . . . . . . . . . . . . 120Chapter 3. Arithmetic functions. . . . . . . . . . . . . . . . . . 891221A note on S(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232On certain inequalities involving the Smarandache function . . . . . . . . . 1243On certain new inequalities and limits for the Smarandache function . . . . 1294On two notes by M. Bencze . . . . . . . . . . . . . . . . . . . . . . . . . . 1375A note on S(n2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386Non-Jensen convexity of S . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397A note on S(n), where n is an even perfect nunber . . . . . . . . . . . . . . 1408On certain generalizations of the Smarandache function . . . . . . . . . . . 1419On an inequality for the Smarandache function . . . . . . . . . . . . . . . 15010The Smarandache function of a set . . . . . . . . . . . . . . . . . . . . . . 1525
11On the Pseudo-Smarandache function . . . . . . . . . . . . . . . . . . . . . 15612On certain inequalities for Z(n) . . . . . . . . . . . . . . . . . . . . . . . . 15913On a dual of the Pseudo-Smarandache function . . . . . . . . . . . . . . . 16114On Certain Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . 16715On a new Smarandache type function . . . . . . . . . . . . . . . . . . . . . 16916On an additive analogue of the function S . . . . . . . . . . . . . . . . . . 17117On the difference of alternate compositions of arithmetic functions . . . . . 17518On multiplicatively deficient and abundant numbers . . . . . . . . . . . . . 17919On values of arithmetical functions at factorials I . . . . . . . . . . . . . . 18220On certain inequalities for σk21Between totients and sum of divisors: the arithmetical function ψ . . . . . 19322A note on certain arithmetic functions . . . . . . . . . . . . . . . . . . . . 21823A generalized Pillai function . . . . . . . . . . . . . . . . . . . . . . . . . . 22224A note on semigroup valued multiplicative functions . . . . . . . . . . . . . 225. . . . . . . . . . . . . . . . . . . . . . . . . 189Chapter 4. Divisibility properties of numbers and functions2271On a divisibility property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2282On a non-divisibility property . . . . . . . . . . . . . . . . . . . . . . . . . 2303On two properties of Euler’s totient . . . . . . . . . . . . . . . . . . . . . . 2324On a conjecture of Smarandache on prime numbers . . . . . . . . . . . . . 2345On consecutive primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2356On Bonse-type inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 2387On certain inequalities for primes . . . . . . . . . . . . . . . . . . . . . . . 2418On certain new conjectures in prime number theory . . . . . . . . . . . . . 2439On certain conjectures by Russo . . . . . . . . . . . . . . . . . . . . . . . . 24510On certain limits related to prime numbers . . . . . . . . . . . . . . . . . . 24711On the least common multiple of the first n positive integers . . . . . . . . 255Chapter 5. Some irrationality results1259An irrationality criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2606
2On the irrationality of certain alternative Smarandache series . . . . . . . . 2633On the Irrationality of Certain Constants Related to the SmarandacheFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2654On the irrationality of et (t Q) . . . . . . . . . . . . . . . . . . . . . . . 2685A transcendental series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2706Certain classes of irrational numbers . . . . . . . . . . . . . . . . . . . . . 2717On the irrationality of cos 2πs (s Q) . . . . . . . . . . . . . . . . . . . . 286Final References288Author Index2947
Chapter 1. Geometric theorems”Recent investigations have made it clear that there exists a very intimate correlationbetween the Theory of numbers and other departments of Mathematics, not excludinggeometry.”(Felix Klein, Evanston Colloquium Lectures, p.58)8
1On Smarandache’s Podaire TheoremLet A0 , B 0 , C 0 be the feet of the altitudes of an acute-angled triangle ABC(A0 BC, B 0 AC, C 0 AB). Let a0 , b0 , b0 denote the sides of the podaire triangleA0 B 0 C 0 . Smarandache’s Podaire theorem [2] (see [1]) states thatXa0 b 0 1X 2a4(1)where a, b, c are the sides of the triangle ABC. Our aim is to improve (1) in the followingform:X1 X 0 21 X 2 1 X 2ab aa a. 31240 0(2)First we need the following auxiliary proposition.Lemma. Let p and p0 denote the semi-perimeters of triangles ABC and A0 B 0 C 0 , respectively. Thenpp0 .2(3)Proof. Since AC 0 b cos A, AB 0 c cos A, we getC 0 B 0 AB 02 AC 02 2AB 0 · AC 0 · cos A a2 cos2 A,so C 0 B 0 a cos A. Similarly one obtainsA0 C 0 b cos B,A0 B 0 c cos C.Thereforep0 1X 0 0 1XRXAB a cos A sin 2A 2R sin A sin B sin C222(where R is the radius of the circumcircle). By a 2R sin A, etc. one hasp0 2Rwhere S area(ABC). By p Y aS ,2RRS(r radius of the incircle) we obtainrrp0 p.R9(4)
Now, Euler’s inequality 2r R gives relation (3).For the proof of (2) we shall apply the standard algebraic inequalities3(xy xz yz) (x y z)2 3(x2 y 2 z 2 ).Now, the proof of (2) runs as follows:X1 X 0 2 1 0 2 1 2 10 0a (2p ) p ab 3333 X 2a4 1X 2a.4Remark. Other properties of the podaire triangle are included in a recent paper ofthe author ([4]), as well as in his monograph [3].Bibliography1. F. Smarandache, Problèmes avec et sans problemes, Ed. Sompress, Fes, Marocco,1983.2. www.gallup.unm.edu/ smarandache3. J. Sándor, Geometric inequalities (Hungarian), Ed. Dacia, Cluj, 1988.4. J. Sándor, Relations between the elements of a triangle and its podaire triangle, Mat.Lapok 9/2000, pp.321-323.10
2On a Generalized Bisector TheoremIn the book [1] by Smarandache (see also [2]) appears the following generalization ofthe well-known bisector theorem.Let AM be a cevian of the triangle which forms the angles u and v with the sides ABand AC, respectively. ThenM B sin vAB ·.ACM C sin u(1)We wish to mention here that relation (1) appears also in my book [3] on page 112,where it is used for a generalization of Steiner’s theorem. Namely, the following resultholds true (see Theorem 25 in page 112):Let AD and AE be two cevians (D, E (BC)) forming angles α, β with the sidesb 90 and α β, thenAB, AC, respectively. If AAB 2BD · BE. CD · CEAC 2(2)Indeed, by applying the area resp. trigonometrical formulas of the area of a triangle,we getA(ABD)AB sin αBD CDA(ACD)AC sin(A α)(i.e. relation (1) with u α, v β α). Similarly one hasBEAB sin(A β) .CEAC sin βThereforeBD · BE CD · CE ABAC 2sin α sin(A β)·.sin β sin(A α)(3)Now, identity (3), by 0 α β 90 and 0 A β A α 90 gives immediatelyrelation (2). This solution appears in [3]. For α β one hasBD · BE CD · CE ABAC 2(4)which is the classical Steiner theorem. When D E, this gives the well known bisectortheorem.11
Bibliography1. F. Smarandache, Proposed problems of Mathematics, vol.II, Kishinev Univ. Press,Kishinev, Problem 61 (pp.41-42), 1997.2. M.L. Perez, htpp/www.gallup.unm.edu/ smarandache/3. J. Sándor, Geometric Inequalities (Hungarian), Ed. Dacia, 1988.12
3Some inequalities for the elements of a triangleIn this paper certain new inequalities for the angles (in radians) and other elementsof a triangle are given. For such inequalities we quote the monographs [2] and [3].x(0 x π) and its first derivative1. Let us consider the function f (x) sin xf 0 (x) 1(sin x x cos x) 0.sin xHence the function f is monotonous nondecreasing on (0, π), so that one can write f (B) f (A) for A B, i.e.AB ,babecause of sin B (1)aband sin A . Then, since B A if b a, (1) implies the2R2RrelationAa , if a b.Bb2. Assume, without loss of generality, that a b c. Then in view of (i),(i)ABC ,abcand consequently A B 0,(a b)ab(b c) B C bc 0,(c a) C A ca 0.Adding these inequalities, we obtainX(a b) A B ab 0,i.e.2(A B C) XA(b c) .aAdding A B C to both sides of this inequality, and by taking into account ofA B C π, and a b c 2s (where s is the semi-perimeter of the triangle) we getXA3π .(ii)a2sThis may be compared with Nedelcu’s inequality (see [3], p.212)XA3π(ii)’ .a4R13
Another inequality of Nedelcu says thatX12s .(ii)”AπrHere r and R represent the radius of the incircle, respectively circumscribed circle ofthe triangle.3. By the arithmetic-geometric inequality we have 1XAABC 3 3.aabc π 3ABC, that is Then, from (ii) and (2) one hasabc2s 3abc2s(iii) .ABCπ4. Clearly, one has 2 2 2 yyxzzx 0,b By a Axc Cz b Bya Ax c Cz(2)or equivalently,z x cax y aby z bc· · · 2xaAybBzcC abc BCCAAB .(3)By using again the A.M.-G.M. inequality, we obtain 1abc 3bca 3.ABCBCCAABThen, on base of (iii), one gets abc6s .πBCCAAB(4)Now (4) and (3) implies thaty z bcz x cA x y ab12s(iv)· · · .xaAybBzcCπ 1 1 1By putting (x, y, z) (s a, s b, s c) or, ,in (iv), we can deduce respectivelya b ccaab12sbc ,A(s a) B(s b) C(s c)πb c c a a b12s ,ABCπwhich were proved in [1].h πi2x, (x 0, , see [3], p.201) in anπ2acute-angled triangle, we can deduce, by using a 2R sin A, etc. that5. By applying Jordan’s inequality sin x 14
(v)Xa12 R.AπBy (ii) and the algebraic inequality (x y z) 1 1 1 x y z 9, clearly, one canobtain the analogous relation (in every triangle)Xa6(v)’ s.Aππ 2 x2sin xfor x (0, π). 2Now, Redheffer’s inequality (see [3], p.228) says thatxπ x2 X3 3sin A , an easy calculation yields the following interesting inequalitySince2 X A33 3 π .(vi)π 2 A24Similarly, without using the inequality on the sum of sin’s one can deduceXaX π 2 A2(vii) 2R.Aπ 2 A2From this other corollaries are obtainable.Bibliography1. S. Arslanagić, D.M. Milosević, Problem 1827, Crux Math. (Canada) 19(1993), 78.2. D.S. Mitrinović et. al., Recent advances in geometric inequalities, Kluwer Acad.Publ. 1989.3. J. Sándor, Geometric inequalities (Hundarian), Ed. Dacia, Cluj, 1988.15
4On a geometric inequality for the medians,bisectors and simedians of an angle of a triangleThe simedian AA2 of a triangle ABC is the symmetrical of the median AA0 to theangle bisector AA1 . By using Steiner’s theorem for the points A1 and A0 , one can writeAB 2A2 B A0 B· .A2 C A0 CAC 2Since A2 B A2 C a, this easily impliesA2 B ac2,b2 c 2A2 C ab2.b2 c 2Applying now Stewart’s theorem to the point B, A2 , C and A:c2 A2 C a · AA22 b2 · A2 B A2 B · A2 C · a;with the notation AA2 sa , the following formula can be deduced:s2a b2 c 2[2(b2 c2 ) a2 ]222(b c )(1)This gives the simedian corresponding to the angle A of a triangle ABC. Let AA0 mabe the median of A. Then, as it is well-known,ma 1p 22(b c2 ) a2 ,2so by (1) one can deduce thatsa 2bcmab2 c2(2)Clearly, this impliessa m a(3)with equality only for b c, i.e. for an isosceles triangle. Let AA1 la be the bisector ofangle A. It is well-known that la ma , but the following refinement holds also true (see[2], p.112).b2 c2ma 1la4bc16(4)
We shall use in what follows this relation, but for the sake of completeness, we give asketch of proof: it is known thatma · la p(p a)(see [2], pp.1001-101), where p a b cdenotes the semiperimeter. Therefore2ma lap(p a) (b c)2(b c)2ma 2 · ,lala4bcp(p a)4bcgiving (4). We have used also the classical formulala 2 pbcp(p a).b cNow, OQ.591, [1] asks for all α 0 such that α αlala 2masaIn view of (2), this can be written equivalently as 1/αla2 f (α) kmakα 1(5)(6) 2bckα 1 1/α Mα (k, 1) is the well-known Hölder mean of argu.Here2b2 c 2ments k and 1. It is known, that Mα is a strictly increasing, continuous function of α,where k andlim Mα α 0 k Mα lim Mα 1α 1(since 0 k 1). Thus f is a strictly decreasing function with values between k· kkand k. For α (0, 1] one hasf (α) f (1) 2k4bc. k 1(b c)2la f (α), i.e. a solution of (6) (and (5)). So, one can saymathat for all α (0, 1], inequality (5) is true for all triangles. Generally speaking, howeverOn view (4) this givesα0 1 is not the greatest value of α with property (5). Clearly, the equationf (α) 17lama(7)
can have at most one solution. If α α0 denotes this solution, then for all α α0la f (α). Here α0 1. Remark that α α0 , relaton (6) is not true, sinceone hasmalaf (α) f (α0 ) .maBibliography1. M. Bencze, OQ.591, Octogon Mathematical Magazine, vol.9, no.1, April 2001, p.670.2. J. Sándor, Geometric inequalities (Hungarian), Editura Dacia, 1988.
József Sándor GEOMETRIC THEOREMS, DIOPHANTINE EQUATIONS, AND ARITHMETIC FUNCTIONS AB/AC (MB/MC)(sin u / sin v) 1/x 1/y 1/z Z(n) is the smallest integer m such that 1 2 m is divisible by n ***** American Research Press
NDOR SiteManager Materials Management Construction Project Manager Materials Compliance Guide Written by Devin Townsend NDOR SiteManager Materials Management Page 3 of 11 1. Introduction and Purpose: This document is intended to be used in conjunction with the existing training materials provided by the
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The formula for the sum of a geometric series can also be written as Sn a 1 1 nr 1 r. A geometric series is the indicated sum of the terms of a geometric sequence. The lists below show some examples of geometric sequences and their corresponding series. Geometric Sequence Geometric Series 3, 9, 27, 81, 243 3 9 27 81 243 16, 4, 1, 1 4, 1 1 6 16 .
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The first term in a geometric sequence is 54, and the 5th term is 2 3. Find an explicit form for the geometric sequence. 19. If 2, , , 54 forms a geometric sequence, find the values of and . 20. Find the explicit form B( J) of a geometric sequence if B(3) B(1) 48 and Ù(3) Ù(1) 9. Lesson 4: Geometric Sequences Unit 7: Sequences S.41
The first term in a geometric sequence is 54, and the 5th term is 2 3. Find an explicit form for the geometric sequence. 19. If 2, , , 54 forms a geometric sequence, find the values of and . 20. Find the explicit form B( J) of a geometric sequence if B(3) B(1) 48 and Ù(3) Ù(1) 9. Lesson 7: Geometric Sequences
been proven to be stable and effective and could significantly improve the geometric accuracy of optical satellite imagery. 2. Geometric Calibration Model and the Method of Calculation 2.1. Rigorous Geometric Imaging Model Establishment of a rigorous geometric imaging model is the first step of on-orbit geometric calibration
